The dominant weight polyhedron conjecture for arbitrary positive-level weights

Let $\mathfrak{g}$ be an untwisted affine Kac–Moody Lie algebra, let $\delta$ be its null root, and let $\lambda\in\mathcal{P}^+$ be a dominant integral weight of positive level. For each proper subset $J\subsetneq I$, let $\widehat b_J(\lambda)$ denote the corresponding vertex candidate of the dominant weight polyhedron $D_\lambda$. The dominant weight polyhedron conjecture. The vertices of $D_\lambda$ are precisely

$$\{\widehat b_J(\lambda)\}_{J\subsetneq I},$$

possibly with repetitions, and

$$D_\lambda=\operatorname{conv}_{\mathbb{Q}}\left(\{\widehat b_J(\lambda)\}_{J\subsetneq I}\right)+\mathbb{Q}_+(-\delta).$$

The preceding proposition proves the analogous decomposition for regular dominant weights. The conjecture extends the vertex description and decomposition to arbitrary positive-level dominant weights, including nonregular ones; the source does not state a resolution.